Paper · Jul 4, 2024

Convergence rate of the -curvature flow

Pak Tung Ho, Sanghoon Lee

math.DGWider flowsarXiv:2407.08757
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Abstract

Carlotto, Chodosh and Rubinstein have studied the convergence rate of the Yamabe flow. Inspired by their result, we study the convergence rate of the -curvature flow in this paper. In particular, we provide an example of a slowly converging -curvature flow in dimension 6, in constrast to the dimension 2 case, where the -curvature flow always converges exponentially.

Comments: All comments welcome! arXiv admin note: substantial text overlap with arXiv:2107.09616, arXiv:2212.04367; substantial text overlap with arXiv:1401.3738 by other authors

Yamabe flowOther curvature flowsLong-time existence & convergence

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence rate of the weighted Yamabe flow

    Pak Tung Ho, Jinwoo Shin and Zetian YanWider flows PDF

    Topics Yamabe flow · Same author Pak Tung Ho · Shared terms “convergence rate”, “Carlotto”, “Rubinstein”

  2. A non-convergence phenomenon for the CR Yamabe flow

    Claudio Afeltra, Andrea Pinamonti and Pak Tung HoWider flows PDF

    Topics Yamabe flow · Same author Pak Tung Ho

  3. Convergence of the fractional Yamabe flow for arbitrary initial energy

    Jingeon An-Lacroix, Hardy Chan and Pak Tung HoWider flows PDF

    Topics Yamabe flow · Same author Pak Tung Ho · Shared terms “inspired”

  4. Global Convergence of the Gursky-Malchiodi -curvature Flow

    Liuwei Gong, Sanghoon Lee and Juncheng WeiWider flows PDF

    Topics Yamabe flow, Other curvature flows · Same author Sanghoon Lee

  5. The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant

    Gilles Carron, Eric Chen and Yi WangWider flows PDF

    Topics Yamabe flow · Shared terms “always converges”, “flow always”

  6. Combinatorial Yamabe flow on hyperbolic bordered surfaces

    Shengyu Li, Xu Xu and Ze ZhouWider flows PDF

    Topics Yamabe flow, Long-time existence & convergence · Shared terms “converges exponentially”

  7. Generalized Yamabe Flows

    Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro VittiWider flows PDF

    Topics Yamabe flow, Long-time existence & convergence

  8. Prescribing hyperbolic bordered surfaces via combinatorial flows

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    Topics Yamabe flow, Long-time existence & convergence, Other curvature flows · Shared terms “converges exponentially”

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Yamabe flow Wider flows The Yamabe flow deforms a metric within its conformal class, driven by scalar curvature. It is a parabolic approach to the Yamabe problem: find a… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.